Nonlinear BoundaryStabilizationfor Timoshenko BeamSystem · [17], for a nonlinear Timoshenko system...
Transcript of Nonlinear BoundaryStabilizationfor Timoshenko BeamSystem · [17], for a nonlinear Timoshenko system...
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Nonlinear Boundary Stabilization for
Timoshenko Beam System
In memorian ao Prof. Silvano B. Menezes
A. J. R. Feitosa ∗, M. L. Oliveira † M. Milla Miranda ‡
abstract:
This paper is concerned with the existence and decay of solutions of the following Timoshenko system:∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
u′′− µ(t)∆u + α1
n∑
i=1
∂v
∂xi
= 0, ∈ Ω× (0,∞),
v′′ −∆v − α2
n∑
i=1
∂u
∂xi
= 0, ∈ Ω× (0,∞),
subject to the nonlinear boundary conditions,∥
∥
∥
∥
∥
∥
∥
u = v = 0 inΓ0 × (0,∞),∂u
∂ν+ h1(x, u′) = 0 in Γ1 × (0,∞),
∂v
∂ν+ h2(x, v′) + σ(x)u = 0 in Γ1 × (0,∞),
and the respective initial conditions at t = 0. Here Ω is a bounded open set of Rn with boundary Γ
constituted by two disjoint parts Γ0 and Γ1 and ν(x) denotes the exterior unit normal vector at x ∈ Γ1.
The functions hi(x, s), (i = 1, 2) are continuous and strongly monotone in s ∈ R.
The existence of solutions of the above problem is obtained by applying the Galerkin method with a
special basis, the compactness method and a result of approximation of continuous functions by Lipschitz
continuous functions due to Strauss. The exponential decay of energy follows by using appropriate
Lyapunov functional and the multiplier method.
Key words and phrases: Timoshenko beam, Galerkin method, Boundary stabilization.
Mathematics Subject classifications: 35L70, 35L20, 35L05
1 introduction
The small vibrations of an elastic beam of length L when are considered the rotatory
inertial and sheared force can be studied by the following system of equations,
∂2u
∂t2(x, t)− cd
∂2u
∂x2(x, t)− cdv(x, t) = 0, 0 < x < L, t ≥ 0
∂2v
∂t2(x, t)− c
∂2v
∂x2(x, t)− c2d
∂u
∂x(x, t) + c2dv(x, t) = 0, 0 < x < L, t ≥ 0
(1.1)
∗Universidade Federal da Paraıba, CCEN-DM, PB, Brasil, [email protected]†Universidade Federal da Paraıba, UFPB, DM, PB, Brasil, ,[email protected]‡Universidade Estadual da Paraıba, DM, Brasil,[email protected]
1
2 Nonlinear system 2005
Completed with the boundary conditions
u(0, t) = 0, v(0, t) = 0,∂u
∂x(L, t) = 0,
∂v
∂x(L, t) = 0, t > 0, (1.2)
and initial conditions
u(x, 0) = u0, v(x, 0) = v0,∂u
∂t(x, 0) = u1(x),
∂v
∂t(x, 0) = v1(x), 0 < x < L (1.3)
Here u(x, t) and v(x, t) denote the transversal displacement and the rotation, respec-
tively, of the point x of the beam at the instant t. In (1.1), c and d represent the
constants:
c =AL2
I1and d =
GI1
EI
where A is the cross sectional area, G is the modulus of elasticity and E is the shear
Young modulus, respectively, of the beam. I, I1 are the axial inertial moment and
polar moment, respectively.
System (1.1) was introduced by Timoshenko [19]. In Tuscnak [20] can be found a
nonlinear version of (1.1). The boundary condition (1.2) denote that the end x = 0 of
the beam remains fixed and the end x = L, built-in, with the boundary conditions
∥
∥
∥
∥
∥
∥
∥
∥
u(0, l) = v(0, l) = 0, t > 0;
cd[
∂u∂x(L, t)− v(L, t)
]
= −δ ∂u∂x(L, t), t > 0, (δ constant);
∂v∂x(L, t) = −τ ∂v
∂t(L, t), t > 0, (τ > 0 constant).
(1.4)
Kim and Renardy [5] studied the existence of solutions of (1.1). Tucsnak [20] obtained
the existence and exponential decay of solutions for this nonlinear version of (1.1) but
with small initial data.
Let Ω be a bounded open set of Rn with boundary Γ constituted by two disjoint
parts Γ0 and Γ1, Γ0
⋂
Γ1= Φ. By ν(x) we represent the exterior unit normal vector at
x ∈ Γ1. A significant generalization of Problem (1.1), (1.3), (1.4) is the following:
∥
∥
∥
∥
∥
∥
∥
∥
∥
u′′
(x, t)− µ(t)∆u(x, t) + α1
n∑
i=1
∂v
∂xi= 0, x ∈ Ω, t > 0
v′′(x, t)−∆v(x, t)− α2
n∑
i=1
∂u
∂xi= 0, x ∈ Ω t > 0
(1.5)
A. J. R. Feitosa , M.L.Oliveira , M.Milla Miranda 3
∥
∥
∥
∥
∥
∥
∥
∥
∥
u(x, t) = 0, v(x, t) = 0 x ∈ Γ0, t > 0;∂u
∂ν(x, t) + h1(x, u
′(x, t)) = 0, x ∈ Γ1 t > 0;
∂v
∂ν(x, t) + h2(, x, v
′(x, t)) + σ(x)u(x, t) = 0, x ∈ Γ1, t > 0;
(1.6)
u(x, 0) = u0(x), v(x, 0) = v0(x), u′(x, 0) = u1(x), v′(x, 0) = v1(x) x ∈ Ω . (1.7)
Here µ(t), σ(x), h1(x, u′(x, s)), h2(x, u
′(x, s)) are real functions defined in t >
0, x ∈ Γ1 and x ∈ R, respectively, and α1, α2 are constants.
In Mota [17] was analyzed the existence and exponential decay of solutions of
Problem (1.5) - (1.7). In this work, the author consider a nonlinear version of (1.5) but
the boundary conditions on Γ1 are linear, i. e.; h1(x, s) = δ1(x)s and h2(x, s) = δ2(x)s.
Of course, the initial data are small.
In the case of wave equation (i. e., when µ = 1 and α = 0 in (1.5)1) with linear
boundary dissipation on Γ1 (i. e.; h(x, s) = δ(x)s), Komornik and Zuazua [7], using
the semigroup theory, showed the existence of solutions. Under the same conditions,
but applying the Galerking method with a special basis, Milla Miranda and Medeiros
[15] obtained similar results. The second method, furthermore to be constructive, has
the advantage of showing the Sobolev space where lies∂u
∂ν.
The above second method has been applied with success to obtain existence of
solutions of divers equations, first, with linear boundary dissipations and then, for
nonlinear boundary dissipations. In the first case, we can mention the papers of Clark et
al. [4], for a coupled system; Araruna and Maciel [1], for the Kirchhoff equation; Mota
[17], for a nonlinear Timoshenko system and Araujo et al. [2], for a beam equation.
In the second case, we cite, among others, the works of Louredo and Milla Miranda
[11],for a coupled system of Klein-Gordon equations; Louredo and Milla Miranda[12],
for a coupled system of Kirchhoff equations and Louredo et al. [13], for a nonlinear
wave equation.
The existence of solutions of the wave equations with a nonlinear boundary dissi-
pations has been obtained, among other, applying the theory of monotone operators
by Zuazua [22], Lasiecka and Tataru [8] and Komornik [6], and applying the Galerkin
method by Vitillaro [21] and Cavalcanti et al. [3]
In all of the above works, the exponential decay of solutions is obtained by applying
a Lyapunov functional and the technique of multipliers, see Komornik and Zuazua [7].
4 Nonlinear system 2005
It is worth emphasizing that the known results in the exponential decay of solution
of the wave equation with nonlinear boundary dissipation where obtained by supposing
that h(s) has a linear behavior in the infinite, i.e.,
d0|s| ≤ |h(s)| ≤ d1|s| ∀s ≥ R (1.8)
R sufficiently large (d0 and d1 positive constants), see Komornik [6] and the references
therein.
In this paper we study the existence and decay of solutions of Problem (1.5) - (1.7).
In the existence of solutions we consider two general functions hi(x, s) (i = 1, 2) which
are continuous and strongly monotone in s, i. e.,
[hi(x, s)− hi(x, r)] ≥ di(s− r)2, ∀s, r ∈ R, x ∈ Γ1, (i = 1, 2)
In this part we apply the Galerkin method with a special basis, the compactness method
and a result of approximation of continuous functions by Lipschitz continuous functions
(see Straus [18]). The choice of the special basis allows us to bound the approximate
solutions (ulm), (vlm) of Problem (1.5) - (1.7) at t = 0. This in turn permits us
to pass to the limit in the nonlinear parts (hi(., u′
lm)), (i = 1, 2). The exponential
decay of energy is obtained for particular hi(x, s) = [m(x)ν(x)]pi(s), (i = 1, 2), where
m(x) = x − x0 and pi(x) is continuous, strongly monotone and satisfies (1.8). In
this part we use an appropriate Lyapunov functional and the multiplier method. It is
important to emphasize that initially we do not know if the sign of the derivative of the
energy E(t) associated to our system is negative, to overcome this difficulty, we add to
it an appropriate functional F (t), so that the derivative of ddt(E+F ) becomes negative
and thus we prove that the energy of the studied system decays at an exponential rate.
Until now we do not know any work where the sign of derivative of the energy of the
system is not known. This is a novelty in our work.
2 Notations and Main Results
Let Ω be a bounded open set of Rn with a C2-boundary Γ constitutedbe two disjoint
parts Γ0, Γ1 with Γ0
⋂
Γ1= Φ and mes(Γ0) > 0, mes(Γ1) > 0. The scalar product and
norm of the real Hilbert space L2(Ω) are denoted by (u, v) and |u|, respectively. By V
is represent tel the Hilbert space.
V = u ∈ H1(Ω); u = 0 in Γ0
A. J. R. Feitosa , M.L.Oliveira , M.Milla Miranda 5
provided with the sorts product and norm
((u, v)) =n
∑
i=1
(∂u
∂xi,∂v
∂xi), ||u|| = ((u, u))2
Let A = −∆ be the self-adjoint operator determined by the triplet V, L2(Ω), ((, ))
(see Lions [10]). Then
D(−∆) = u ∈ V ∩H2(Ω);∂u
∂ν= 0 on Γ1
In order to state the result on the existence of solutions, we introduce the necessary
hypotheses. Consider functions
hi ∈ C0(R, L∞(Γ1)), hi(x, 0) = 0 a. e., x ∈ Γ1 (2.1)
(i = 1, 2) which are strongly monotone in the second variable, i.e.,
[hi(x, s)− hi(x, r)] ≥ di(s− r)2, ∀s, r ∈ R (2.2)
a.e. x ∈ Γ1 where di are positive constants (i = 1, 2). Also consider
µ ∈ W1, 1loc (0, ∞), µ(t) ≥ ν0 > 0, ∀t ∈ [0,∞), (ν0 constant) (2.3)
and
σ ∈ W 1, ∞(Γ1) (2.4)
Theorem 2.1 Assume hypotheses (2.1) - (2.4). consider two numbers α1 6= 0 and
α2 6= 0 and vectors
u0 ∈ D(−∆) ∩H10 (Ω), v0 ∈ D(−∆), and u1, v1 ∈ H1
0 (Ω) (2.5)
Then there exists a pair of functions u, v in the class∥
∥
∥
∥
∥
∥
∥
u, v ∈ L∞
loc(0, ∞; V )
u′, v′ ∈ L∞
loc(0, ∞; V )
u′′
, v′′
∈ L∞
loc(0, ∞; L2(Ω))
(2.6)
such that u, v satisfy the system
u′′
(x, t)−mu(t)∆u(x, t) + α1
n∑
i=1
∂v
∂xi= 0, in L∞(0, ∞; L2) (2.7)
v′′(x, t)−∆v(x, t)− α2
n∑
i=1
∂u
∂xi= 0, in L∞(0, ∞; L2) (2.8)
6 Nonlinear system 2005
the boundary conditions
∂u
∂ν+ h1(., u
′) = 0 in L∞
loc(0, ∞; L1(Γ1)) (2.9)
∂u
∂ν+ h2(., v
′) + σu = 0 in L∞
loc(0, ∞; L1(Γ1)) (2.10)
and the initial conditions
u(0) = u0, v(0) = v0, u′(0) = u1, v′(0) = v1 (2.11)
In what follows, we introduce the notations and hypotheses to state the result on
the decay of solutions. We will use the notations.
|u| ≤M ||u||, ||u||L2(Γ1) ≤ N ||u||, ∀u ∈ V (2.12)
Consider the function m(x) = x− x0, x ∈ Rn (x0 a fixed vector de R
n ). Assume
that there exist x0 ∈ Rn such that
Γ0 = x ∈ Γ; m(x).ν(x) ≤ 0 Γ1 = x ∈ Γ; m(x).ν(x) > 0 (2.13)
Use the notations,
R(x0) = max||m(x)||Rn ; x ∈ Ω, 0 < τ0 = minm(x).ν(x); x ∈ Γ1 (2.14)
Assume the
h1(x, r) =[
m(x).ν(x)]
p1(s) h2(x, r) =[
m(x).ν(x)]
p2(s) (2.15)
where pi (i = 1, 2) satisfy
∥
∥
∥
∥
∥
∥
∥
∥
pi ∈ C0(R), pi(0) = 0[
pi(s)− pi(r)]
(s− r) ≥ bi(s− r)2, ∀s, r ∈ R
|pi(s)| ≤ li|s|, s ∈ R
(2.16)
were bi, and Li are positive constants.
We consider two real numbers α1 > 0, and α2 > 0. Introduce the following
notations
A = 2(n− 1)M
µ1
2
0
+ 2(n− 1)Mα1
α2+ 4
R(x0)
µ1
2
+ 4R(x0)α1
α2(2.17)
A. J. R. Feitosa , M.L.Oliveira , M.Milla Miranda 7
P1 = 4(n− 1)2nM2
µ0+ 16R2(x0)
n
µ0(2.18)
∥
∥
∥
∥
∥
∥
∥
∥
∥
P2 = 4(n− 1)2||
n∑
i=1
νi||2L∞(Γ1)
N4
µ0+
2N2
τ0µ0||
n∑
i=1
νi||2 + 16R2(x0)
n
µ0
(2.19)
S1 = 4(n− 1)2µ(0)R(x0)L21N
2 + µ(0)R2(x0)L22 + 1 (2.20)
S2 = 4(n− 1)2R(x0)L22N
2 + 2R2(x0)L22 + 1 (2.21)
With respect to positive real numbers α1, and α2, we assume the following
hypotheses:
α1α2 ≤µ0
64nM2and P1α
21 + P2α
22 ≤
7
8(2.22)
We consider a positive functions σ(x) given by
σ(x) = α2
(
n∑
i=1
νi(x))
(2.23)
We take three real numbers ǫ1, ǫ2, and η satisfying
0 < ǫ1 ≤1
4A, 0 < ǫ2 ≤ min
µ0b1
S1
,(α1
α2)b2
S2
(2.24)
and
0 < η ≤ minǫ1, ǫ2 (2.25)
Introduction the energy
E(t) =1
2
|u′(t)|2 +α1
α2|v′(t)|2 + µ(t)||u(t)||2 +
α1
α2||v(t)||2
, t ≥ 0 (2.26)
8 Nonlinear system 2005
Theorem 2.2 Assume hypotheses (2.13), (2.15), (2.16), (2.22) and (2.23). Assume
also that
µ′(t) ≤ 0, a.e. t ∈ (0,∞) (2.27)
Then the pair of solutions u, v given in Theorem (2.1) satisfy
E(t) ≤ 3E(0)e−2
3ηt, ∀t > 0 (2.28)
where η > 0 was defined in (2.25)
3 Existence of Solutions
Before proving Theorem 2.2, we need of some previous results.
Lemma 3.1 Let h(x, s) be a function satisfying the hypotheses (2.1) and (2.2) with
d0 > 0. Then there exists sequence (hl) of vectors of C0(R;L∞(Γ1)) satisfying the
following conditions:
(i) hl(x, 0) = 0 a.e. x ∈ Γ1;
(ii)[
hl(x, s)− hl(x, r)]
(s− r) ≥ d0(s− r)2, ∀s, r ∈ R and a.e. ∈ Γ1
(iii) For anyl ∈ N there exists a function cl ∈ L∞(Γ1) satisfying
|hl(x, s)− hl(x, s)| ≤ cl|s− r|, ∀s, r ∈ R an a.e. in Γ1
(iv) (hl) converges to h uniformly in bounded sets of R, a.e. x ∈ Γ1
Lemma 3.2 Let T > 0 be a real number. Consider the sequence (wl) of vectors of
L2(0, T ;H−1
2 (Γ1))∩L1(0, T ;L1(Γ1)) and w ∈ L2(0, T ;H−
1
2 (Γ1)) and χ ∈ L1(0, T ;L1(Γ1)
such that
(i) wl → w weak in L2(0, T ;H−1
2 (Γ1))
(ii) wl → χ in L1(0, T ;L1(Γ1))
then, w = χ
The proof of Lemma 3.1 can be found in Strauss [18] and Lemma 3.2, in Louredo and
Milla Miranda [11].
A. J. R. Feitosa , M.L.Oliveira , M.Milla Miranda 9
3.1 Proof of Theorem 2.1
Let (h1l) and (h2l) be two sequences in the conditions of Lemma 3.1 that approximate
h1 and h2, respectively, consider two sequences (u1l ) and (v1l ) of vectors of C∞
0 (Ω) such
that
u1l → u1 and v1l → v1 in H10 (Ω) (3.1)
Note that∥
∥
∥
∥
∥
∂u0
∂ν+ h1l(., u
1l ) = 0 on Γ1 ∀l
∂v0
∂ν+ h2l(., v
1l ) + σu0 = 0 on Γ1 ∀l
(3.2)
Now, we fix l ∈ N and construct a basis wl1, w
,2w
l3... of V ∩ H2(Ω) such that
u0, v0, u1l , v1l belong to the subspace [w
l1, w
,2w
l3, w
l4] spanned by the vectors wl
1, w,2w
l3, w
l4.
Let Vm = [wl1, w
l2, w
l3, ...w
lm] be the subspace of V ∩H2(Ω) spanned by wl
1, wl2, w
l3, ...w
lm.
Approximated Problem: We find an approximate solution ulm, vlm of Problem
(1.5) - (1.7) belonging to Vm, i. e.
ulm(t) =n
∑
i=1
gjlm(t)wlj, vlm(t) =
n∑
i=1
hjlm(t)wlj
and ulm, vlm is a solution of the system,
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
(u′′
lm, ϕ) + µ((ulm, ϕ)) + µ∫
Γ1h1l(., u
′
lm)ϕdΓ1+
α1(
n∑
i=1
∂vlm
∂xi, ϕ) = 0, ∀ϕ ∈ Vm
(v′′
lm, ψ) + ((vlm, ψ)) +∫
Γ1h2l(., v
′
lm)ψdΓ1 +∫
Γ1σulmψdΓ1−
α2(
n∑
i=1
∂ulm
∂xi, ψ) = 0, ∀ψ ∈ Vm
ulm(0) = u0, vlm(0) = v0, u′lm(0) = u1l , v′
lm(0) = v1l
(3.3)
The above finite-dimensional system has a solution ulm, vlm defined in [0, tlm). The
following estimates allow us to extend this solutions to the interval [0,∞].
3.2 Estimates I
Consider ϕ = u′lm and ψ = v′lm in (3.2)1 and (3.2)2, respectively, we obtain
10 Nonlinear system 2005
d
dt
[1
2|u′lm|
2 +µ
2||ulm||
2]
+ µ
∫
Γ1
h1(., u′
lm)u′
lmdΓ1+
α1(
n∑
i=1
∂vlm
∂xi, u′lm) =
µ′
2||ulm||
2(3.4)
and
d
dt
[1
2|v′lm|
2 +1
2||vlm||
2]
+
∫
Γ1
h2(., v′
lm)v′
lmdΓ1+
∫
Γ1
σulmv′
lmdΓ1 − α2(
n∑
i=1
∂ulm
∂xi, v′lm) = 0
(3.5)
Introduce the notation
Elm(t) =1
2|u′lm(t)|
2 +1
2|v′lm(t)|
2 +µ
2||ulm(t)||
2 + ||vlm(t)||2 (3.6)
we add the both sides of (3.4) and (3.5) and use hypothesis (2.3) on µ and Lemma 3.1,
part(ii), applied to h1l, h2l we have
ddtElm + µ0d1
∫
Γ1
u′2lmdΓ1 + d2
∫
Γ1
v′2lmdΓ1 ≤µ′
2||ulm||
2−
α1
(
n∑
i=1
∂vlm
∂xi, u′lm
)
−
∫
Γ1
σulmv′
lmdΓ1 + α2
(
n∑
i=1
∂ulm
∂xi, v′lm
) (3.7)
We find
∣
∣
∣
∣
α1
( n∑
i=1
∂vlm
∂xi, u′lm
)∣
∣
∣
∣
≤ α21n(1
2||vlm||
2)
+1
2|u′lm|
2
Similarly
∣
∣
∣
∣
α2
( n∑
i=1
∂ulm
∂xi, v′lm
)∣
∣
∣
∣
≤ α22
n
µ0
(µ
2||ulm||
2)
+1
2|v′lm|
2
Also,
∣
∣
∣
∣
∫
Γ1
σulmv′
lmdΓ1
∣
∣
∣
∣
≤N2
µ0d2||σ||2L∞(Γ1)
(µ
2||ulm||
2)
+d2
2||v′||2L2(Γ1)
Taking into account the last three inequations in (3.7), derive
A. J. R. Feitosa , M.L.Oliveira , M.Milla Miranda 11
d
dtElm + µ0d1
∫
Γ1
u′2lmdΓ1 +d2
2
∫
Γ1
v′2lmdΓ1 ≤(
|µ′|+K)
Elm
where K is the constant
K = α21 + α2
2
n
µ0
+N2
µ0d2||σ||2L∞(Γ1)
So, integrating the preceding inequality on [0, t), t < tlm, we find
Elm + µ0d1
∫ t
0
∫
Γ1
u′2lmdΓ1ds+d2
2
∫ t
0
∫
Γ1
v′2lmdΓ1ds ≤ Elm(0) +
∫ t
0
(
|µ′|+K)
Elmds
Convergence (3.1) yield
Elm(0) ≤1
2|u1|2 +
1
2|v1|2 +
µ(0)
2||u0||2 + ||v0||2 + 1 = L0, ∀l ≥ l0
Thus, the last two inequalities and Gronwall Lemma provide
12|u′lm(t)|
2 + 12|v′lm(t)|
2 + µ
2||ulm(t)||
2 + ||vlm(t)||2 + µ0d1
∫ t
0
∫
Γ1
u′2lmdΓ1ds+
d22
∫ t
0
∫
Γ1
v′2lmdΓ1ds ≤ L0exp
∫ t
0
(|µ′|+K)dt = C(T ), ∀t ∈ [0, T ], ∀l ≥ l0
(3.8)
where the constat C(T ) > 0 is independent of l ≥ l0 and m. So
(ulm) and (vlm) are bounded in L∞
loc(0,∞;V )
(u′lm) and (v′lm) are bounded in L∞
loc(0,∞;L2(Ω))
(u′lm) and (v′lm) are bounded in L∞
loc(0,∞;L2(Γ1)))
(3.9)
3.3 Estimates II
Differentiating with respect to t the approximate equation (3.3)1 and making ϕ = u′′
lm
in the resulting expression, we obtain
d
dt
[1
2|u
′′
lm|2 +
µ
2||u
′
lm||2]
+ µ′((ulm, u′′
lm)) + µ′
∫
Γ1
h1l(., u′
lm)u′′
lmdΓ1 +
µ
∫
Γ1
h1l(., u′
lm)(u′′
lm)2dΓ1 + α1
(
n∑
i=1
∂v′lm∂xi
, u′′
lm
)
=µ′
2||u′lm||
2
12 Nonlinear system 2005
Considering α = µ′
µu
′′
lm in (3.3)1 we get
µ′((ulm, u′′
lm)) + µ′
∫
Γ1
h1l(., u′
lm)u′′
lmdΓ1 = −µ′
µ|u′lm|
2 −µ′
µ
(
α1
n∑
i=1
∂vlm
∂xi, u
′′
lm
)
Combining the last two equations, we final
d
dt
[1
2|u
′′
lm|2 +
µ
2||u
′
lm||2]
+ µ
∫
Γ1
h′1l(., u′
lm)(u′′
lm)2dΓ1 =
µ′
2||u′lm||
2 +
µ′
µ|u
′′
lm|2 + α1
µ′
µ
(
n∑
i=1
∂vlm
∂xi, u
′′
lm
)
− α1
(
n∑
i=1
∂v′lm∂xi
, u′′
lm
)
In similar way, approximate equation (3.3)2 provide,
d
dt
[1
2|v
′′
lm|2 +
1
2||v
′
lm||2]
+
∫
Γ1
h′2l(., v′
lm)(v′′
lm)2dΓ1 =
α2
(
n∑
i=1
∂u′lm∂xi
, v′′
lm
)
−
∫
Γ1σ(x)u′
lmv′′
dΓ
Introduce the notation
E∗
lm(t) =1
2|u
′′
lm|2 +
1
2|v
′′
lm|2 +
µ
2||u
′
lm||2 +
1
2|v
′
lm|2, t ≥ 0
Adding the both sides of the las two equations and using hypothesis (2.3) on µ an
Lemma 3.1, part (ii), applied to h1l, h2l, we get,
d
dtE∗
lm + µd1
∫
Γ1
(u′′
lm)2dΓ1 + d2
∫
Γ1
(v′′
lm)2dΓ1 ≤
µ′
2||u′lm||
2 +
α1µ′
µ
(
n∑
i=1
∂vlm
∂xi, u
′′
lm
)
− α1
(
n∑
i=1
∂v′lm∂xi
, u′′
lm
)
+ α2
(
n∑
i=1
∂u′lm∂xi
, v′′
lm
)
− (3.10)
∫
Γ1
σu′lmv′′
lmdΓ1
we have
A. J. R. Feitosa , M.L.Oliveira , M.Milla Miranda 13
•∣
∣
∣α1µ′
µ
(
n∑
i=1
∂vlm
∂xi, u
′′
lm
)∣
∣
∣≤
(α1)2
µ20
n|µ′|(1
2|vlm|
2) + |µ′|(1
2)|u
′′
lm|2
•∣
∣
∣α1
(
n∑
i=1
∂v′lm∂xi
, u′′
lm
)∣
∣
∣≤ (α1)
2n(1
2||v′lm||
2) +1
2|u
′′
lm|2
•∣
∣
∣α2
(
n∑
i=1
∂u′lm∂xi
, v′′
lm
)∣
∣
∣≤α22
µ0n(µ
2)||u′lm||
2 +1
2|v
′′
lm|2
•∣
∣
∣
∫
Γ1
σu′lmv′′
dΓ1
∣
∣
∣≤
N2
d2µ0||σ||2L∞(Γ1)(
µ
2||u2lm||
2) +1
2||v
′′
lm||2L2(Γ1)
from the last four inequalities in (3.10) and using the boundedness (3.8) for ||vlm||2 we
have
d
dtE∗
lm + µ0d1
∫
Γ1
(u′′
lm)2dΓ1 +
d2
2
∫
Γ1
(v′′
lm)2dΓ1 ≤ K1(T )|µ
′|+[
K2|µ′|+K3
]
E∗
lm (3.11)
where
K1(T ) =α21
µ0nC(T ), K2 =
1
µ0+ 1, K3 = α2
1n+α22
µ0n +
N2
d2µ0||σ||L∞(Γ1) + 2 (3.12)
Integrate both sides of (3.11) on [0, t], 0 < t ≤ T , we find
E∗
lm(t) ≤ E∗
lm(0) +K1(T )
∫ T
0
|µ′|dt+
∫ T
0
[K2|µ′|+ k3]E
∗
lmds (3.13)
We will obtain a second estimate if we bound E∗
lm(0). This is the key point of the
proof of theorem 2.1. The boundedness will follow by the choice of the special basis of
V ∩H2(Ω).
In fact, if we make t = 0 in the approximate equations (3.3)1 and (3.3)2 and consider
ϕ = u′′lm and ψ = u′′lm we have
|u′′
lm(0)|2 + µ(0)((u0, u
′′
lm(0))) + µ(0)
∫
Γ1
h1l(., u1l )u
′′
lmdΓ1 + α1
(
n∑
i=1
∂v0
∂xi, u
′′
lm(0))
= 0
(3.14)
and
|v′′
lm(0)|2 + µ(0)((v0, v
′′
lm(0))) +
∫
Γ1
h2l(., v1l )v
′′
lmdΓ1 +
∫
Γ1
σu0u′′
lmdΓ1 +
α1
(
n∑
i=1
∂v0
∂xi, u
′′
lm(0))
= 0. (3.15)
14 Nonlinear system 2005
The Gauss Theorem and the equalities (3.2) provide
µ(0)((u0, u′′
lm(0))) + µ(0)
∫
Γ1
h1l(., u1l )u
′′
lm(0)dΓ1 =
µ(0)[
(−∆u0, u′′
lm(0)) +
∫
Γ1
(∂u0
∂µ+ h1l(., u1l ))u
′′
lm(0)dΓ1
]
=
µ(0)(−∆u0, u′′
lm(0)).
Taking into account in the last two equations (3.14) and (3.15), we get
|u′′
lm(0)|2 + µ(0)(−∆u0, u
′′
lm(0)) + α1
(
n∑
i=1
∂v0
∂xi, u
′′
lm(0))
= 0
and
|v′′
lm(0)|2 + µ(0)(−∆v0, v
′′
lm(0))− α2
(
n∑
i=1
∂u0
∂xi, v
′′
lm(0))
= 0.
So
|u′′
lm(0)|2 ≤ µ(0)|∆u0|+ |α1|n
1
2 ||v0|| = a1 (3.16)
and
|v′′
lm(0)|2 ≤ µ(0)|∆v0|+ |α2|n
1
2 ||u0|| = a2. (3.17)
Therefore, the last two boundedness and convergence (3.1) provide
E∗
lm(0) ≤a212
+a222
+µ(0)
2||u1||2 +
1
2||v1||2 + 1 = a3, ∀l ≥ l0. (3.18)
The inequalities (3.16) and (3.18) and Gronwall Lemma yields
12|u
′′
lm(t)|2 + 1
2|v
′′
lm(t)|2 + µ(t)
2||u′lm(t)||
2 + 12||v′lm(t)||
2 + µ0d1
∫
Γ1
(u′′
lm)2dΓ1
+d22
∫
Γ1
(v′′
lm)2dΓ1 ≤
[
a3 +K1(T )
∫ t
0
|µ′|dt]
exp
∫ t
0
[
K2|µ′|+ k3
]
dt = C1(T )
∀t ∈ [0, T ], l ≥ l0
(3.19)
where C1(T ) > 0 is a constant independent of l ≥ l0 and m.
Thus∥
∥
∥
∥
∥
∥
∥
(u′lm) and (v′lm) are bounded in L∞
loc(0,∞;V )
(u′′
lm) and (v′′
lm) are bounded in L∞
loc(0,∞;L2(Ω))
(u′′
lm) and (v′′
lm) are bounded in L2loc(0,∞;L2(Γ1))
(3.20)
A. J. R. Feitosa , M.L.Oliveira , M.Milla Miranda 15
The boundedness (3.9) and (3.20) provide two subsequences of (ulm) and (vlm), still
denoted by ulm and vlm, and two function ul and vl such that
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
ulm → ul and vlm → vl werk star in L∞
loc(0,∞;V )
u′lm → u′l and v′
lm → v′l werk star in L∞
loc(0,∞;V )
u′′
lm → u′′
l and v′′
lm → v′′
l werk star in L∞
loc(0,∞;L2(Ω))
u′lm → u′l and v′
lm → v′l werk in L2loc(0,∞;L2(Γ1))
u′′
lm → u′′
l and v′′
lm → v′′
l werk in L2loc(0,∞;L2(Γ1))
. (3.21)
3.4 Passage to the Limit in m
We analyze the nonlinear terms on the boundary Γ1. Let T > 0 be a real number. By
convergence (3.21)2 and (3.21)5, the compact embedding of H1
2 (Γ1) in L2(Γ1) and the
Aubin-Lions Theorem [9 ], give us
u′
lm → u′l in L2loc(0,∞;L2(Γ1))
Lemma 3.1, part (iv), provide
∫
Γ1
|h1l(., u′
lm)− h1l(., u′
l)|2dΓ1 ≤ ||cl||L∞(Γ1)
∫
Γ1
|u′lm − u′l|2dΓ1
These two results yield
h1l(., u′
lm) → h1l(., u′
l) in L2loc(0,∞;L2(Γ1))
Then by a diagonal process, we obtain
h1l(., u′
lm) → h1l(., u′
l) in L2loc(0,∞;L2(Γ1)) (3.22)
In a similar way, we find
h2l(., v′
lm) → h2l(., v′
l) in L2loc(0,∞;L2(Γ1)) (3.23)
We take the limit in m of system (3.3). Then by convergence (3.21), (3.22), (3.23)
and noting that Vm is dense in V , we obtain
16 Nonlinear system 2005
∫
∞
0
(u′′
l , ϕ)θdt+
∫
∞
0
µ((ul, ϕ))θdt+
∫
∞
0
∫
Γ1
µh1l(., u′
l)ϕθdΓ1dt+
∫
∞
0
α1
(
n∑
i=1
∂vl
∂xi, ϕ
)
dt = 0 ; ∀ϕ ∈ V, ∀θ ∈ D(0,∞)(3.24)
and
∫
∞
0
(v′′
l , ψ)θdt+
∫
∞
0
((vl, ψ))θdt+
∫
∞
0
∫
Γ1
h2l(., v′
l)ψθdΓ1dt+
∫
∞
0
∫
Γ1
σulψθ −
∫
∞
0
α2
(
n∑
i=1
∂ul
∂xi, ψ
)
dt = 0 ; ∀ψ ∈ V, ∀θ ∈ D(0,∞)(3.25)
Considering ϕ, ψ in D(Ω) in the preceding equations and noting the regularity of
ul, vl given in (refa32), we get
u′′
l − µ∆ul + α1
n∑
i=1
∂vl
∂xi= 0 in L∞
loc(0,∞;L2(Ω))
v′′
l −∆vl − α2
n∑
i=1
∂ul
∂xi= 0 in L∞
loc(0,∞;L2(Ω))
(3.26)
This implies that ∆ul, ∆vl belongs to L∞
loc(0,∞;L2(Ω)) and ul and vl belong to
L∞
lon(0,∞;V ), we find∂ul
∂xi,∂vl
∂xiin L2
loc(0,∞;H−1
2 (Γ1)), see [15].
Multiplying both sides of equation (3.26) by ϕθ and ψθ with ϕ, ψ in V and
θ ∈ D(0,∞), using the green formulae and preceding regularity, we obtain
∫
∞
0
(u′′
l , ϕ)θdt+
∫
∞
0
µ((ul, ϕ))θdt−
∫
∞
0
< µ∂ul
∂ν, ϕ > θdt+
∫
∞
0
α1
(
n∑
i=1
∂vl
∂xi, ϕ
)
θdt(3.27)
and
∫
∞
0
(v′′
l , ψ)θdt+
∫
∞
0
((vl, ψ))θdt−
∫
∞
0
<∂vl
∂ν, ψ > θdt+
∫
∞
0
α2
(
n∑
i=1
∂ul
∂xi, ψ
)
θdt(3.28)
A. J. R. Feitosa , M.L.Oliveira , M.Milla Miranda 17
where < . , . > is the duality pairing between H−1
2 (Γ1) and H1
2 (Γ1) . Comparing
equations (3.27) and (3.24) with (3.28) and (3.25), using the regularity of h1l(., u′
l) and
h2l(., v′
l) given in (3.22) and (3.23), respectively, we have
∂ul
∂ν+ h1l(., u
′
l) = 0 in L∞
loc(0,∞, L2(Γ1))
∂vl
∂ν+ h2l(., v
′
l) + σul = 0 in L∞
loc(0,∞, L2(Γ1))(3.29)
3.5 Passage to the Limit in l
As the boundedness (3.8) and (3.19) are independent of l ≥ l0 and m, we obtain
analogous convergence to (3.7), i. e., there are functions u and v such that
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
∥
ul → u and vl → v werk star in L∞
loc(0,∞;V )
u′l → u′ and v′l → v′l werk star in L∞
loc(0,∞;V )
u′′
l → u′′
and v′′
l → v′′
werk star in L∞
loc(0,∞;L2(Ω))
u′l → u′ and v′l → v′ werk in L2loc(0,∞;L2(Γ1))
u′′
l → u′′
and v′′
l → v′′
werk in L2loc(0,∞;L2(Γ1))
(3.30)
These convergence allow us to pass to the limit in (3.24) and (3.25). So for ϕ, ψ in
D(Ω), we obtain
∥
∥
∥
∥
∥
∥
∥
∥
∥
u′′
− µ∆u+ α1
n∑
i=1
∂v
∂xi= 0 L∞
loc(0,∞, L2(Γ1))
v′′
− µ∆v + α2
n∑
i=1
∂u
∂xi= 0 L∞
loc(0,∞, L2(Γ1))
(3.31)
In what follow, we analyze the equation (3.29). Let T > 0 be a fixed real number. The
convergence (3.30)2 yields
u′l → u′ wark in L2(Γ1)
This, the compact immersion of H1
2 (Γ1) in L2(Γ1) and the Aubin - Lions Theorem,
give us
u′l → u′ in L2(0, T : L2(Γ1))
which implies
u′l(x, t) → u′(x, t) a.e. x ∈ Γ1, l ∈ (0, T ).
18 Nonlinear system 2005
Analogously,
v′l(x, t) → v′(x, t) a.e. x ∈ Γ1, l ∈ (0, T ).
Fix (x, t) ∈ Γ1×(0, T ). The last convergence implies that the set u′l(x, t), v′l(x, t); l ≥
l0 is a bounded set of R. This and Lemma 3.1, part (iv), on the uniformly convergence
of h1l(x, u′(x, t)) and h2l(x, u
′(x, t)), provide
h1l(x, u′
l(x, t)) → h1(x, u′(x, t)) in a.e. Γ1 × (0, T )
h2l(x, v′
l(x, t)) → h2(x, v′(x, t)) in Γ1 × (0, T )
(3.32)
We take the scalar product of L2(Ω) on both sides of equation (3.26)1 with u′l and
integrate on [0, T ] to obtain
∫ T
0
∫
Γ1
µh1l(., u′
l)dΓ1dt = −1
2|u′l(T )|
2 +1
2|u′l(0)|
2 −µ(T )
2||ul(T )||
2 +µ(0)
2||u0||2 −
∫ T
0
α1
(
n∑
i=1
∂vl
∂xi, u′l
)
+
∫ T
0
µ′
2||ul||
2dt
By estimate (3.8) we find that each term of the second member of the preceding
expression can be bound by a constant C3(T ). Thus
∫ T
0
∫
Γ1
h1l(., u′
l)u′
ldΓ1dt ≤C4(T )
µ0(3.33)
In a similar way, we get from (3.26)2 that
∫ T
0
∫
Γ1
h2l(., v′
l)v′
ldΓ1dt ≤ C5(T ) (3.34)
The constants C4(T ) and C5(T ) are independent of l ≥ l0. The results (3.32) -
(3.34) allow us to apply the Strauss Theorem [18] to obtain
h1l(., u′
l) → h1(., u′) in L1(Γ1 × (0, T ))
h2l(., v′
l) → h2(., v′) in L1(Γ1 × (0, T ))
(3.35)
On the other hand, by convergence (3.30) and equation (3.26)1 we deduce that
uj → u weak in L2(0, T ;V ) and ∆ul → ∆u weak in L2(0, T ;L2(Ω)).
A. J. R. Feitosa , M.L.Oliveira , M.Milla Miranda 19
Therefore
∂ul
∂ν→
∂u
∂νweak inL2(o, T ;H−
1
2 (Γ1))
See [15]. By equation (3.29)1 and convergence (3.35)1, we get
∂ul
∂ν= −h1l(., u
′
l) → h1(., u′) in L1(0, T ;L1(Γ1))
The last convergence and Lemma 3.2 provide
∂ul
∂ν+ h1(., u
′
l) = 0 in L1(0, T ;L1(Γ1))
Then by a diagonal process, we obtain
∂ul
∂ν+ h1(., u
′
l) = 0 in L1loc(0,∞;L1(Γ1)) (3.36)
In a similar way, we deduce
∂vl
∂ν+ h2(., v
′
l) + σu = 0 in L1loc(0,∞;L1(Γ1)) (3.37)
The initial condition (2.11) is obtained from (3.3)3 and the estimates (3.21), (3.30).
With the above part, the equations (3.31), (3.35), (3.37) and the estimates (3.30)
we get the proof of theorem 2.1.
4 Decay of Solutions
Before proving the theorem 2.2, we introduce some previous results.
Proposition 4.1 Let h : R → R be a Lipschitz continuous function. If u ∈ H1
2 (Γ1),
then h(u) ∈ H1
2 (Γ1) and the map h : H1
2 (Γ1) → L1
2 (Γ1) is continuous.
Let (pil) be the sequence of Lipschitz continuous functions given in Lema 3.1 that
approximate pi (i = 1, 2). Note that u′l ∈ L∞
loc(0,∞;H1
2 (Γ1)) (see(3.21)2) and Γ is of
class C2. The proposition 4.1 implies that
(m · ν)p1l(u′
l) ∈ L∞
loc(0,∞;H1
2 (Γ1)).
20 Nonlinear system 2005
This and (3.26)1 provide
∂ul
∂ν= −(m · ν)p1l(u
′
l) = gl ∈ L∞
loc(0,∞;H1
2 (Γ1))
Also,the equation (3.26)1 implies that ∆ul ∈ L∞
loc(0,∞;L2(Ω)).
Thus ul(t) is the solution of the following elliptic problem
−∆ul(t) = ft in Ω (fl(t) inL2(Ω))
ul(t) = 0 on Γ0
∂ul(t)
∂ν= gl on Γ1 (gl(t)) ∈ H
1
2 (Γ1)
By regularity of elliptic problems we have
ul ∈ L∞
loc(0,∞;V ∩H2(Ω)) (4.1)
(see [15]). Similarly,
vl ∈ L∞
loc(0,∞;V ∩H2(Ω)). (4.2)
The regularity (4.1) allows us to obtain the following identities
∥
∥
∥
∥
∥
∥
∥
(∆ul, m · ∇ul) = (n− 2)||ul||2 −
∫
Γ
(m · ν)|∇ul|2dΓ + 2
∫
Γ
∂ul
∂ν(m · ∇ul)dΓ
2(u′l, m · ∇u′l) = −n|u′l|2 +
∫
Γ
(m · ν)(u′)2dΓ(4.3)
(see [7]). Symilarly
∥
∥
∥
∥
∥
∥
∥
(∆vl, m · ∇vl) = (n− 2)||vl||2 −
∫
Γ
(m · ν)|∇vl|2dΓ + 2
∫
Γ
∂ul
∂ν(m · ∇ul)dΓ
2(v′l, m · ∇v′l) = −n|v′l|2 +
∫
Γ
(m · ν)(v′)2dΓ.(4.4)
4.1 Proof of Theorem 2.2
We will prove the inequality (2.28) of theorem 2.2 for solutions ul, vl given by theorem
2.1 with h1(x, s) = (m(x) · ν(x))p1(s) and h2(x, s) = (m(x) · ν)p2(s). The result
follows by taking the infimum limit on both sides of the obtained inequality and using
convergence (3.30).
A. J. R. Feitosa , M.L.Oliveira , M.Milla Miranda 21
In order to facilitate the writing, we will omit the sub-index l of the diverse expres-
sions.
Introduce the notation
E(t) =1
2
(
|u′(t)|2 +α1
α2|σ′(t)|2 + µ(t)||u(t)||2 +
α1
α2||v(t)||2
)
t ≥ 0 (4.5)
By similar computations made to obtain (3.4), we deduce from (3.26)1 and (3.26)2
that after multiplyingα1
α2
by (3.26) we obtain
dE
dt=u′
2||u||2 − µ
∫
Γ
(m, ν)p2(v′)v′dΓ− α1
(
n∑
i=1
∂v
∂xi, u′
)
−
α1
α2
∫
Γ1
(m, ν)p2(v′)v′dΓ−
α1
α2
∫
Γ1
σuv′dΓ + α1
(
n∑
i=1
∂u
∂xi, v′
)
(4.6)
By Gauss Theorem, we have
α1
(
n∑
i=1
∂v′
∂xi, u
)
= α1
∫
Γ1uv′
(
n∑
i=1
νi
)
dΓ1 − α1
(
n∑
i=1
∂u
∂xi, v′
)
that implies
d
dt
(
α1
n∑
i=1
∂v
∂xi, u
)
= α1
∫
Γ1
uv′(
n∑
i=1
νi
)
dΓ1 − α1
(
n∑
i=1
∂u
∂xi, v′
)
+ α1
(
n∑
i=1
∂v
∂xi, u′
)
.
Therefore
−α1
(
n∑
i=1
∂v
∂xi, u′
)
= −α1
(
n∑
i=1
∂u
∂xi, v′
)
+ α1
∫
Γ1
uv′(
n∑
i=1
νi
)
dΓ1 −d
dt
(
α1
n∑
i=1
∂v
∂xi, u
)
Combining this equality with (4.6)and canceling similar terms with opposite signs,
we obtain
dE
dt+d
dt
(
α1
n∑
i=1
∂v
∂xi, u
)
=u′
2||u||2 − µ
∫
Γ1
(m, ν)p1(u′)u′Γ1 −
α1
α1
∫
Γ1
(m, ν)p2(v′)v′dΓ1 −
∫
Γ1
[α1
α2σ − α1
(
n∑
i=1
νi
)]
uv′dΓ1
Then the hypothesis (2.23) implies that
dE
dt+d
dt
(
α1
n∑
i=1
∂v
∂xi, u
)
=u′
2||u||2 − µ
∫
Γ1
(m, ν)p1(u′)u′Γ1 −
α1
α1
∫
Γ1
(m, ν)p2(v′)v′dΓ1 ≤ 0
22 Nonlinear system 2005
Using the notation
F (t) = α1
(
n∑
i=1
∂v
∂xi(t), u(t)
)
, t ≥ 0 (4.7)
we obtain
d
dt
(
E + F)
=u′
2||u||2 − µ
∫
Γ1
(m, ν)p1(u′)u′Γ1 −
α1
α1
∫
Γ1
(m, ν)p2(v′)v′dΓ1. (4.8)
The above equality provide bounded solutions on [0,∞]. In order to obtain the
decay of solutions, we introduce the functional
G(t) = (n− 1)(u′, u) + (n− 1)(v′, v) + 2(u′, m.∇u) + 2(v′, m.∇v). (4.9)
4.2 Boundedness of F an G
We have
|F | ≤ 2(α1.α2
µ0
n)
1
2
ME (4.10)
where M is the constant introduced in (2.12). Then we obtain
•∣
∣
∣(n− 1)(u′, u)| ≤ 2(n− 1)
M
µ1
2
0
E
•∣
∣
∣(n− 1)(v′, v)
∣
∣
∣≤ 2(n− 1)M α1
α2E
•∣
∣
∣2(u′, m.∇u)
∣
∣
∣≤ 4
R(x0)
µ012
E where R(x0) was introduced in (2.14)
•∣
∣
∣2(v′, m.∇v)
∣
∣
∣≤ 4R(x0)
α1
α2
E
Thus
|G| ≤ AE (4.11)
Where the constant A was introduced in (2.17). From (4.10), (4.11) and ε, it follows
that
|F + εG| ≤[
2(α1α2n
µ0
)1
2M + εA]
E
For the particular α1 α2 satisfying hypothesis (2.22), we have
|F + ε1G| ≤1
2E, 0 ≤ ε1 ≤
1
4s
So1
2E(t) ≤ E(t) + F (t) + ε1G(t) ≤
2
3E(t), t ≥ 0. (4.12)
A. J. R. Feitosa , M.L.Oliveira , M.Milla Miranda 23
4.3 Boundedness of dG
dt
We have
dG
dt= (n− 1)(u
′′
, u) + (n− 1)|u′|2 + (n− 1)(v′′
, v) + (n− 1)|v′|2+
2(u′′
, m.∇u) + 2(u′, m.∇u′) + 2(v′′
, m.∇v) + 2(v′, m.∇v′) =
I1 + (n− 1)|u′|2 + I2 + (n− 1)|v′|2 + I3 + I4 + I5 + I6
(4.13)
• By equation (3.26)1, we find
I1 = (n− 1)(µ∆u, u)− α1(n− 1)(
n∑
i=1
∂v
∂xi, u)
and by equation (3.29)1, we also find
I1 = −(n− 1)µ||u||2 − (n− 1)µ
∫
Γ1
(m.ν)h1(u′)udγ1 − (n− 1)α1
(
n∑
i=1
∂v
∂xi, u
)
• In a similar way, by (3.26)2 and (3.29)2, we derive,
I2 = −(n− 1)||v||2 − (n− 1)
∫
Γ1
(m.ν)h2(v′)vdΓ1 − (n− 1)
∫
Γ1
(σu)vdΓ1 +
(n− 1)α2
(
n∑
i=1
∂u
∂xi, v)
• By equation (3.26)1 and identity (4.3), we get
I3 = µ(n− 1)||u||2 − (n− 1)
∫
Γ1
(m.ν)|∇u|2dΓ1 + 2µ
∫
Γ1
∂u
∂xi(m.∇u)dΓ1 −
2α1
(
n∑
i=1
∂v
∂xi, m.∇u
)
I4 = −|u′|2 +
∫
Γ
(m.ν)u1
2dΓ1.
• In a similar way, by (3.26)2 and (4.3), we find
I5 = µ(n− 1)||v||2 −
∫
Γ1
(m.ν)|∇v|2dΓ1 + 2
∫
Γ1
∂v
∂xi(m.∇v)dΓ1 +
2α2
(
n∑
i=1
∂u
∂xi, m.∇v
)
24 Nonlinear system 2005
I6 = −n|v′|2 +
∫
Γ1
(m.ν)(v′)2dΓ1.
Taking into account the last four equalities in (4.13) and canceling the terms with
opposite signs, we have
dG
dt= −|u′|2 − |v′|2 − µ||u||2 − ||v||2 − (n− 1)µ
∫
Γ
(m.µ)h1(u′)udΓ1−
(n− 1)α1
(
n∑
i=1
∂v
∂xi, u
)
− (n− 1)
∫
Γ1
(m.ν)h2(v′)vdΓ1 − (n− 1)
∫
Γ1
σuvdΓ1+
(n− 1)α2
(
n∑
i=1
∂u
∂xi, v)
− µ
∫
Γ1
(m.ν)|∇v|2dΓ + 2µ
∫
Γ∂u
∂ν(m.∇u)dΓ−
2α1
(
n∑
i=1
∂v
∂xi, m.∇u
)
−
∫
Γ1
(m.ν)|∇v|2dΓ1 + 2
∫
Γ1
∂v
∂xi(m.∇v)dΓ1+
2α2
(
n∑
i=1
∂u
∂xi, m.∇v
)
+
∫
Γ1
(m.ν)(u′)2dΓ1 +
∫
Γ1
(m.ν)(v′)2dΓ1 =
−|u′|2 − |v′|2 − µ||u||2 − ||v||2 +
11∑
k=1
Jk +
∫
Γ1
(m.ν)(u′)2dΓ+
∫
Γ1
(m.ν)(v′)2dΓ
(4.14)
we also have,
• |J1| ≤ 4(n− 1)2µ(0)R(x0)L21N
2
∫
Γ1
(m.ν)(u′)2dΓ1 +1
16µ||u||2
where the constants Li (i = 1, 2) were introduced in hypothesis (2.16)
• |J2| ≤ 4(n− 1)2nM2α21
µ0
(µ||u||2) +1
16||v||2
• |J3| ≤ 4(n− 1)2R(x0)L22N
2
∫
Γ1
(m.ν)(v′)2dΓ +1
16||v||2
The hypothesis (2.23) provides
• |J4| ≤ 4(n− 1)2∣
∣
∣
∣
∣
∣
n∑
i=1
ν∣
∣
∣
∣
∣
∣
L2∞(Γ1)
N4
µ0
α22(µ||u||
2) +1
16||v||2
• |J5| ≤ 4(n− 1)2nM2α2
µ0
(µ||u||2) +1
16||v||2
A. J. R. Feitosa , M.L.Oliveira , M.Milla Miranda 25
Observing that |∇u|2 = (∂u∂ν)2 in Γ0, we find
• J6 = −µ
∫
Γ0
(m.ν)(∂u
∂ν)2dΓ0 − µ
∫
Γ0
(m.ν)|∇u|2dΓ1 (4.15)
noting that ∇u = ν. ∂u∂xi
on Γ0, we obtain
J7 = 2µ
∫
Γ0
(m.ν)(∂u
∂ν)2dΓ0 + 2µ
∫
Γ0
∂u
∂ν(m.∇u)dΓ1
On the other hand, using equations (3.26)1 and the hypothesis (2.14), (2.27), we
get
∣
∣
∣2µ
∫
Γ0
∂u
∂ν(m.∇u)dΓ1
∣
∣
∣≤ µ(0)R2(x0)
∫
Γ1
1
m.ν(∂u
∂xi)2dΓ1 +
∫
Γ0
(m.ν)|∇u|2dΓ1
µ(0)R2(x0)L21
∫
Γ1
(m.ν)(u′)2dΓ1 + µ
∫
Γ1
(m.ν)|∇u|2dΓ1
So
J7 ≤ 2µ
∫
Γ0
(m.ν)(∂u
∂ν)2dΓ0 + µ(0)R2(x0)L2
1
∫
Γ1
(m.ν)(u′)2dΓ1+
µ
∫
Γ1
(m.ν)|∇u|2dΓ1
(4.16)
Therefore, after adding (4.15) and (4.16), reducing similar terms, canceling similar
terms with opposite signs and noting that m.ν ≤ 0 on Γ0, we obtain
J6 + J7 ≤ µ(0)R2(x0)L21
∫
Γ1
(m.ν)(u′)2dΓ1
• |J8| ≤ 16R2(x0) nµ0α21(µ||u||
2) + 116||v||2
• In a similar way as in (4.16), we find
J9 + J10 ≤ R2(x0)
∫
Γ1
1
m.ν[−(mν)h2(v
′)− σu]2dΓ1 ≤
2R2(x0)L22
∫
Γ1
(m.ν)(v′)2dΓ1 + 2α22
∫
Γ1
1
m.ν
(
n∑
i=1
νi
)2
u2dΓ1,(4.17)
we obtain
2α22
∫
Γ1
1
m.ν
(
n∑
i=1
νi
)2
u2dΓ1 ≤ 2||
n∑
i=1
νi||2L∞(Γ1)
N2
τ0µ0α2(µ||u||
2)
26 Nonlinear system 2005
where the constant τ0 was introduced in (2.14). The preceding two inequalities
provides
J9 + J10 ≤ 2R2(x0)L22
∫
Γ1
(m.ν)(v′)2dΓ1 + 2||n
∑
i=1
νi||2L∞(Γ1)
N2
τ0µ0
(µ||u||2)
• |J11| ≤ 16R2(x0) nµ0α22(µ||u||
2) + 116||v||2
Taking into account the above boundedness for Ji, (i = 1, ..., 11) in (4.14) and
using notations introduced in (2.18)- (2.21) we obtain,
dG
dt≤ −|u′|2 − |v′|2 − µ||u||2 − ||v||2 + p1α
21(µ||u||
2) + p2α22(µ||u||
2) +
S1
∫
Γ1
(m.ν)(u′)2dΓ1 + S2
∫
Γ1
(m.ν)(v′)2dΓ1 +1
16µ||u||2 +
3
8||v||2
This implies
dG
dt≤ −|u′|2 − |v′|2 − µ||u||2 − ||v||2 + p1α
21(µ||u||
2) + (p1α1 + p2α2)(µ||u||2) +
S1
∫
Γ1
(m.ν)(u′)2dΓ1 + S2
∫
Γ1
(m.ν)(v′)2dΓ1
The hypothesis (2.22) provide 1516
−[
p1α1 + p2α22] ≥
12. Then
dG
dt≤ −|u′|2 − |v′|2 − µ||u||2 − ||v||2 + S1
∫
Γ1
(m.ν)(u′)2dΓ1 + S2
∫
Γ1
(m.ν)(v′)2dΓ1
We note that 12α1
α2≤ 1
2or −1
2< −1
2α1
α2for all α1 > 0 and α2 > 0. Thus
dG
dt≤ −E + S1
∫
Γ1
(m.ν)(u′)2dΓ1 +
∫
Γ1
(m.ν)(v′)2dΓ1 (4.18)
In the sequel, we conclude the proof of Theorem 2.2. By (4.8), (4.17), hypothesis
(2.16) and for ǫ > 0, we have
d
dt(E + F + ǫG) ≤
µ′
2||u||2 − µ0b1
∫
Γ1
(m.ν)(u′)2dγ1 −
α1b2
α2
∫
Γ1
(m.ν)(v′)2dΓ1 − ǫE + ǫS1
∫
Γ1
(m.ν)(u′)2dΓ1 (4.19)
ǫS2
∫
Γ1
(m.ν)(v′)2dΓ1
A. J. R. Feitosa , M.L.Oliveira , M.Milla Miranda 27
Choosing ǫ2 > 0 in conditions (2.24) we find
d
dt(E + F + ǫ2G) ≤ −ǫ2E
taking η > 0 in conditions (2.25) and using (4.12),we get
E(t) + F (t) + ηG(t) ≤ e−2
3η[
E(o) + F (0) + ηF (0)]
Then (4.12) implies that
E(t) ≤ 3e−2
3ηE(0), ∀t ≥ 0
the proof is completed.
Acknowledgement 1 The author M. L. Oliveira acknowledges the support of Na-
tional Institute of Science and Technology of Mathematics INCT-Mat and CAPES and
CNPq/Brazil.
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